Get three points randomly on a circle and connect them to draw a triangle. What's the probability for this triangle to include the center of the circle?

The answer was said to be 0.25 but my calculation shows the answer is .

Here is how I calculated. Please let me know if you observe an error. Thanks

Step1: Take a random point(A) on the circle. The probability P1=1.Step2: Take the second point B on the circle and say the center is point O. The degree of AOB is X and 0<X<= pi. The probability of getting this point is P2= 2 * X/ (2pi) = X/piStep3: To get the point C making triangle ABC to include the center, we need to extend the line AO and BO respectively and intersect on the circle at A' & B'. Point C must fall on the curve A'B' and the probability of this is P3=X/(2pi).Step4: P = P1*P2*P3 = X^2 / (2 pi^2). Do an integral

Get three points randomly on a circle and connect them to draw a triangle. What's the probability for this triangle to include the center of the circle?

The answer was said to be 0.25 but my calculation shows the answer is .

Here is how I calculated. Please let me know if you observe an error. Thanks

Step1: Take a random point(A) on the circle. The probability P1=1.Step2: Take the second point B on the circle and say the center is point O. The degree of AOB is X and 0<X<= pi. The probability of getting this point is P2= 2 * X/ (2pi) = X/piStep3: To get the point C making triangle ABC to include the center, we need to extend the line AO and BO respectively and intersect on the circle at A' & B'. Point C must fall on the curve A'B' and the probability of this is P3=X/(2pi).Step4: P = P1*P2*P3 = X^2 / (2 pi^2). Do an integral

You are on the right track! With the first point fixed, the second point will be at an angle uniformly distributed in [0, pi] with respect to the center. The probability of a third point landing such that the triangle contains the center for a given angle is just , as you surmised by extending the chords.

So integrating the distribution and normalizing it (since it is 0 to pi instead of 0 to 1) gives